It has been known for more than a century that in two-dimensional (planar) Euler flow, vorticity is conserved along streamlines. In three-dimensions, however, no such result has been established, and this is primarily due to the vortex stretching term in the equation of motion. The vorticity persistence theorem is herein extended to three dimensions. It states that in any Euler flow, all components of the vorticity tensor of a streamline coordinate system that are normal to the streamline direction are conserved along streamlines. This extension is accomplished with the aid of a mathematical simplification of the vorticity equation derived for arbitrary coordinate systems. What remains of the nonlinear convective terms in the vorticity equation, after the mathematical simplification, is the Lie derivative of the vorticity tensor with respect to fluid velocity. A coordinate-independent temporal derivative is defined which, when set to zero, expresses either the continuity or vorticity equation (excluding the viscous term), depending upon the argument supplied to it.